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A TEXT-BOOK

OP

DIFEEBENTIAL CALCULUS

A TEXT-BOOK

OF

DIFFERENTIAL CALCULUS

WITH NUMEROUS WORKED OUT EXAMPLES

BY

GANESH^^RASAD

B.A. (Cantab.), D.Sc. (Allahabad)

MEMBER OP THE LONDON MATHEMATICAL. SOCIETY, OF THE DEUTSCHE

MATHEMATIKER-VEREINIGUNG, OP THE CTRCOLO MATEMATICO DI PALERMO, ETC.

FELLOW OF THE UNIVERSITY OF ALLAHABAD

AND PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, BENARE8

LONGMANS, GEEEN, AND CO.

39 PATERNOSTER ROW, LONDON

NEW YORK, BOMBAY, AND CALCUTTA

1909

All rights reserved

OfV f

^

PREFACE.

In this work it has been my aim to lay before students a

strictly rigorous and, at the same time, simple exposition of

the Differential Calculus and its chief applications. The

present volume is intended for beginners and is so designed

as to meet the requirements of Part I. of the Cambridge

Mathematical Tripos Examination, and of the Examinations

for the B.A. and B.Sc. degrees of Indian Universities.

The chief characteristics of the present work may be

indicated as follows : â€” (1) The fundamental principles of the

Differential Calculus have been based on a purely arith-

metical foundation. Thus, the various theorems have been

carefully enunciated and their proofs have been made quite

independent of geometrical intuition. In this connection,

I may specially mention the chapters on Bolle's Theorem

and Taylor's Theorem, Maxima and Minima, and Indeter-

minate Forms. (2) Almost every article is followed by

worked out examples, specially suited for illustrating the

article. There are also numerous exercises in every chapter.

(3) A special chapter deals with curve-tracing and the im-

portant properties of the best- known curves. (4) The order

in which the chapters are arranged is intended to enable the

beginner to study the simple geometrical applications of the

Differential Calculus immediately after he has learnt differen-

tiation. (5) The miscellaneous notes A and B are intended

r- O t i* f> O

vi PREFACE

to give the ambitious student a glimpse of the modern

researches in the Differential Calculus.

This volume is based on my experience in teaching the

elements of the Differential Calculus to a large number of

pupils. It is, therefore, throughout of an elementary

character. But, as certain parts of it may be found difficult

by beginners, they have been marked with an asterisk and

may be omitted in a first reading.

A few words may be said here about Chapter I. In a

mathematical book, which professes to be rigorous in its

treatment, it is essential that the definitions be carefully

worded. This has been done in the present work. It is,

however, possible that, for this reason, Chapter I. may be

found heavy reading by some students. To them my advice

is this : If you do not grasp the full meaning of a definition

in a first reading, leave it and after reading Chapters II. and

III. come back to Chapter I., and then you will understand

the definition better.

In writing the present volume, I have derived much help

from two books, viz. the excellent little manual, ' Calcolo

Differenziale ' of Professor Ernesto Pascal, and Todhunter's

1 Treatise on the Differential Calculus.' For the historical notes

in Chapter VIII., as well as for most of the examples on that

chapter, I am indebted to Professor Gino Loria's ' Spezielle

algebraische und transscendente ebene Kurven.' Of the re-

maining examples in this volume, a large number are common

to all English text-books on the subject, some are original and

the others are taken from the Tripos Examinations of recent

years ; in the case of the more important examples belonging

to the last category, the sources are cited in the text.

To ray friend and former pupil, Mr. Lakshmi Narayan,

M. A., Professor of Mathematics at the Central Hindu College,

PEE FACE vii

Benares, I am much indebted for some valuable suggestions

and for assistance in revising the proof-sheets. I am also

indebted to my friend Dr. S. C. Bagchi, B.A. (Cantab.), LL.D.

(Dublin), Principal of the University Law College of Calcutta,

whose criticisms and suggestions, relating to Chapters V.,

VIII., and XL, have materially enhanced the usefulness of the

present volume.

G. PBASAD.

Benares, July 1909.

CONTENTS.

[N.B. â€” The portions marked with an asterisk may be omitted in a first reading.]

CHAPTEE I.

Definitions.

ART. PAGE

1. Variable 1

2. Function 1

3. Limit 2

4. Continuity 4

5. Differential coefficient 5

*Examples on Chapter 1 6

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

"

CHAPTER II.

Standard Forms.

Introductory

Four important limits

Differential coefficient of x H

Differential coefficient of a x

Differential coefficient of sin x

Differential coefficient of cos x

Differential coefficient of tan x

Differential coefficient of cot x

Differential coefficient of sec x

Differential coefficient of cosec x

Differential coefficient of vers x

Differential coefficient of log a x

Differential coefficient of sin -1 x

Differential coefficient of cos -1 x

Differential coefficient of tan -1 x and cot -1 x

Differential coefficient of sec -1 x and cosec -1 x

Differential coefficient of vers -1 x

Table of results to be committed to memory

Examples on Chapter II

8

9

11

12

13

14

14

14

15

15

15

16

17

18

19

19

20

21

22

x CONTENTS

CHAPTER in.

Fundamental Rules for Differentiation.

ART. PAGE

24. Constant 24

25. Product of constant and function 24

26. Sum of two functions 24

27. Product of two functions 25

28. Product of more than two functions 25

29. Quotient of two functions 27

30. Function of a function 28

31. Inverse functions , . . 28

32. The form \4>{x)}W> 29

Examples on Chapter III. 30

CHAPTER IV.

Tangents and Normals.

33. Tangent : Definition and Cartesian equation .

34.

35.

36.

37.

38.

39.

40.'

41.

42.

43.

44.

45.'

46.

47.

ds

v- sv

Note on the equation ^r

Normal : Definition and Cartesian equation

Cartesian subtangent and subnormal ....

Polar coordinates. Angle between tangent and radius vector

ds /â€ž, . /dr\ 2

\de)

rÂ»+

Note on the equation â€”-= . /

Polar subtangent and subnormal

Perpendicular from pole on tangent .

Pedal equation

Inversion. Pedal curves. Polar reciprocals

Examples on Chapter IV

35

36

39

40

41

42

43

44

45

46

47

CHAPTER V.

Asymptotes.

Definition of asymptote 52

General rule for finding asymptotes from Cartesian equation . . 53

Parallel asymptotes 56

Asymptotes by inspection 57

Note on curvilinear asymptotes 58

General rule for finding asymptotes from polar equation ... 59

Note on circular asymptotes 59

Examples on Chapter V 60

CHAPTER VI.

Curvature.

Centre of curvature : Definition and Cartesian coordinates

Radius of curvature : Definition and formulae .

63

66

STENTS xi

ART. PAOT

48. Circle and chord of curvature 70

49. Evolute 71

50. Concavity and convexity. Point of inflexion 73

Examples on Chapter VI 75

CHAPTER VII.

- Envelopes.

51. Family of curves 78

52. Definition of envelope 78

53. Rule for finding the envelope of a family of straight lines . . 7&

54. General rule 81

Examples on Chapter VII. 82

â™¦CHAPTER VIII.

Curve Tracing. Properties of Special Curves.

55. Introductory 84

56. Rules for Cartesian equations 84

Note on multiple points 85

57. Rules for polar equations 87

58. Parabola. Ellipse. Hyperbola 89

59. Semi-cubical parabola. Cissoid. Folium 91

60. Lemniscate. Cardioid. Conchoid 92

61. Cycloid. Catenary. Tractrix . . 93

62. Logarithmic spiral. Archimedian spiral. Sine spiral ... 95

Examples on Chapter VIII. . . â– 96

CHAPTER IX.

Successive Differentiation.

63. Definitions 99.

64. Standard results 100

65. Leibnitz's theorem 102

66. Use of partial fractions. Recurrence formulas 103

Examples on Chapter IX 107

CHAPTER X.

Rolle's Theorem and Taylor's Theorem.

67. Rolle's theorem. The theorem of the mean value . . . .110'

68. Taylor's development in finite form 112

69. Taylor's theorem. Maclaurin's theorem 115

*Note on contact of curves . 117

Examples on Chapter X H^

xii CONTENTS

CHAPTER XI.

Maxima and Minima.

ART. PA'

70. Definitions 121

71.* Two theorems ". 122

72. General rule for finding maxima and minima 125

Examples on Chapter XI ; 127

CHAPTER XII.

Indeterminate Forms.

73. Introductory 128

74. Cauchy's theorem. The fundamental form - 130

75. The Forms ~ , x Â°o co _ co oÂ°, 1Â°Â°, coo 132

CO

Note on compound indeterminate forms 135

Note on infinitesimals and infinities 137

Examples on Chapter XII. 138

MISCELLANEOUS NOTES.

A. Weierstrass's function 140

B. Rolle's theorem and Taylor's theorem , 143

C. Partial differentiation 146

\J-

DIFFERENTIAL CALCULUS.

CHAPTEE I.

DEFINITIONS.

1. Variable. Let # be a symbol which takes successively every

numerical value from a given number a to another given number /3.

Then x is called a variable and the totality of the values of x

constitutes the domain of x.

We will represent the domain of x by the symbol (a, fi).

Note. If k be a number, it will be convenient to use the symbol | k \

to denote the absolute value of k, i.e., the value of k without regard to its

sign. Thus

I -2 I = I 2-1 = 2.

2. Function. By a function of x, denned for a given domain,

is understood a quantity which has a single and definite value for

every value of x in its domain.

We generally denote functions of x by such symbols as f(x),

*(*), F(x), f(x).

EXAMPLES.

1. # 2 , 2 X , sin x are functions of x whatever be the domain

of x. But sin -1 x cannot be a function of x for such a domain

as (2, 3).

2. The temperature curve at a certain place is y = 80 + 10 sin x. If the

temperatures recorded are all different, the highest and lowest temperatures

being 90Â° and 80Â° respectively, find the domain of x.

3. For the domain (0, 1), a function may be defined by saying that it is

x 11

zero or - according asÂ£C = 0or->#> , n having the values 1 2, 3, etc.

n nâ€” n+l

2 Â»>V DIFFERENTIAL CALCULUS

4. For .the -domain (#, -1)[, & function f(x) may be defined by saying that

f(x) is 2 or 3 according as x is rational or irrational.

3. Limit. A and a being both finite, A is said to be the

limit of f(x) for #=a if, for any number 8, however small but

greater than zero, there exists a corresponding number c>0

such that

x having every value such that

0<

<8,

<â‚¬.

Note 1. This definition may be expressed in a different form, viz. : A is

said to be the limit of f(x) for x = a, if f(x) differs from A by less than any

assigned quantity, however small, when x has any value sufficiently near to a.

We will use the notation __ f(x) to denote the limit of f(x)

for x=a.

f(x) is said to be oo, if, for any positive number N however large, there

x = a

exists a corresponding number e > such that f(x) > N, x having every value

such that

0< 05-a <â‚¬.

f(x) = A, if, for any number 5, however small but greater than zero,

there exists a corresponding number N>0 such that

U-/(a?) I <8,

x having every value greater than N.

li

Definitions similar to the above hold for the cases when fix)- -oo,

x = a

lim f(x) = A lita /(*)=oo,etc.

Xâ€” â€” QO X = QO

Note 2. The notion of limit, on which Differential Calculus is based, is

not so unfamiliar to the beginner as he might at first imagine. For, in his

algebraic studies he must have become acquainted with this notion in connec-

tion with the sums of infinite series. For example, what is meant by saying

that 2 is the sum of the series

i + 2+ 2 \+23 + â€¢ â– â€¢ toinfinit y ?

Nothing but this : 2 is the limit to which S,â€ž the sum to n terms, tends as

n is made greater and greater.

DEFINITIONS

EXAMPLES.

1. **"" x 2 is a 2 . For, take any number Â£ however small, but

Lim

greater than zero. Now, if

#â€” a

<Â«,

where

< 1 ; hence

x=a + 6e,

x + a=2a + 0e,

and, consequently,

Therefore, since

x + a J <2

+ *.

x 2 â€”a

2 =(x-

-a)(x+a),

x 2 â€”a 2

<e

x+a <el% a

Therefore, if e is such that

.(,

a 1 + Â£ )Â±S,

x 2 -a 2 1 <a

for every value of x satisfying the condition

xâ€”a

<â‚¬.

And such a value of e is any positive number which is equal to or

less than the positive root of

tU I a I + A-S=0;

for, as the product of the roots of this equation is â€”2, a negative

quantity, one of the roots is positive and the other negative.

2. In the last example, prove that, when 8 is taken to be â€” , e may be

taken to be

2 | a | 10"

3. Prove that hm cosz = l.

x =

4. If _ f(x) exists and is finite, it follows that, for any number 5, how-

b2

4 DIFFEBENTIAL CALCULUS

ever small but greater than zero, there exists a corresponding number â‚¬ >

such that

|/(z,)-/(x 2 )| <5

for every pair x v x 2 satisfying the conditions

0< a?,â€” a < â‚¬ , 0< \ x 2 â€” a\ <â‚¬.

5. Prove that im cos is non-existent. Let x, and x 2 be respectively

x = x

equal to â€” and ; r- t n being an integer. Then

* 2mr (2n + l)?r

cos â€” cos = 2,

x x x 2

however large n may be. Hence it follows that in * cos is non-existent.

x = x

For, if this limit existed, according to the preceding example, it would be

possible to find a value of ft so large that

cos cos â€” < 5

even when 5 < 2.

6. Prove that im rt , is non-existent.

x = L

2 + e x

4. Continuity. A function f(x) is said to be continuous for

x=a, if im J(x) exists, is finite and equals f(a).

x=a

EXAMPLES.

1. x 1 is continuous for #=a. For, x 2 exists and equals a 2 .

x=a ^

2. A function f(x) is defined by saying that it equals 1 or

e~*\ according as x is zero or different from zero. This function

is discontinuous for #=0. For, e x ' 1 is zero and is not equal

to /(0).

3. If f(x) be continuous for x = 0, prove that /(0) must be zero, when

f(x) = x sin - for values of x different from zero.

4. What are the points of discontinuity of the function given in Ex. 3 of

Art. 2 ?

DEFINITIONS 5

5. Differential Coefficient. By the differential coefficient of a

function f{x) for x=a is understood

lim /(s)-/(a) ^ lim /(a + ft)-/(a)

#=a # - a ' ' 7&=0 /&

The differential coefficient of f(x) for #=& will be denoted by the

symbol f(a) ; the differential coefficient, considered as a function

df(x)

of x, will be generally represented by -} orf'(x).

ctx

Note 1. The process of finding/' (x) is called differentiating f(x).

Note 2. For the geometrical meaning of f'(x), see Art. 33, Note 1.

Note 3. The beginner should not think that â€¢} ' means the ratio of

ax

df(x) to dx. To think so would be as wrong as to think that sin x means the

product of sin and x. Just as sin is meaningless, so are df(x) and dx. As

has been already stated, the symbol *} ' stands for

dx

lim f(x + h) -f(x)

h = Q h

This notation was introduced by Leibnitz in 1676.

EXAMPLES.

1. Find the differential coefficient of x 6 for x=-l.

Here

xâ€”1 xâ€”1

Therefore

/'(1)= Km /(*)-Al) = lim (a .* + s+l)=3.

x=l xâ€”1 X=I

2. Find *&.

dx

3. A function f(x) equals zero or x cos -, according

zero or different from zero. Prove that/'(0

Here

rio)= lim SMSM

J v ' x=0 x

lim 1

#=0 x

But, by Ex. 5, Art. 3, lm * cos - is non-existent. Therefore /'(())

x=0 x J v '

is non-existent.

as x is

x

stan-(l).

6 DIFFEBENTIAL CALCULUS

4. A function f(x) is denned by saying that it equals 0, x or - x according

as a: is 0, > or <0. Prove that/'(0) does not exist.

j

5. Find the differential coefficient of x 3 for x = 0.

6. If f(x) has the same value whatever x may be, prove that

/'(*)-0.

Here

f'tmX- Hm / ( a? + ft )-/( a? ) = lira 9.

1 v ; fc=0 & *=0 ft

â™¦Examples on Chapter I.

1. If /(*)Â»- h â„¢ tan" 1 ( x \ prove that f(x) is equal to 0, 1 or -1

according as x is 0, > or < 0.

2. Trace the curve

y= l[m

3. If /(a?) = , prove that f(x) is or 1 according as a; is or

different from 0.

4. Give the graph of

lim (l + simro;)' 1 â€” 1

y a Â» * ,

n- co (1 + sinirx) w + 1

5. If <Â£>(#) s - â€” â€” p prove that the limit to which <f>(sin n ! *#) tends

when the integer n is made larger and larger, is or 1 according as x is

rational or irrational.

6. A function f(x) is denned by saying that it equals 0. or sin (- )

\sin xj

according as x is, or is not, a multiple of Â». Find the points of discontinuity

of /(*).

7. Trace the curve

y* lim (simp)*- 1 .

n = co

8. A function f(x) is denned by saying that it equals or sin / \

l Sin xj

according as x is either zero or a submultiple of , or neither of these. Prove

IT

that there are an infinite number of points of discontinuity of f(x) between

a and |3, where a < < 3.

DEFINITIONS 7

9. If f(x) = or according as x is zero or different from zero, trace

l-e x

the curve y =f{x).

10. Trace the curve

li m x 2n sin - +x 2

find the values of y at the points x Â». Â± 1, and discuss whether 2/ is continuous

at these points.

[Math. Tripos, 1901.]

. 1

sin ~

11. A function f(x) is defined by saying that it equals or Â£ according

log x*

as x is zero or different from zero. Has f(x) a differential coefficient for x = ?

12. In each of the following cases discuss the question of the existence of

the differential coefficient for x = a : â€”

_ _i

(i) f(x) = e ( * -a)a whenx^a,

/(#) = when x = a.

(ii) /(#) = (# -a) 2 cos when xÂ±a,

x-a '

/(#) s when sc = a.

(iii) /(x) = <c - a when #>a,

/(ic) = a-cc when x < a.

(iv) f(x) = (x â€” a) cos when #=j=a,

/(#) = when x = a.

[Calcutta Univ., 1906.]

CHAPTEE II.

STANDARD FORMS.

6. Introductory. It is the object of the present chapter to

investigate and tabulate the results of differentiating the simple

elementary functions, viz., x n , a x , sin x, cos x, tan x, cot x, sec x,

cosec x, vers x } \og a x, sin -1 x, cos -1 x, tan -1 x, cot -1 x, sec -1 x,

cosec -1 x, vers -1 x.

It will be seen later on that, by means of certain rules to be

given in Chapter III. and a knowledge of the standard forms of

the present chapter, most of the ordinary functions can be easily

differentiated.

Throughout the book we shall always consider the inverse

functions to be so denned that

â€” -j<sin -1 # _<-, CKcos -1 #Â£7r,

A . A

- 2 ^tan *x < 2 ,- 2 ^_cot l *<^

Oj^sec -1 a? <Â»,â€” ^cosec" 1 #_S^,

0<yQr$~ l x<j.

Note. Throughout this book we shall take for granted the truth of the

following theorem : In general,

1Z { *Â« * m) . Â£ { *i - ** } â– x t { :;g } â– Â£ { <^h }

are respectively equal to

lim

lim . , , , lim . ,, lim . , * lim . , v scrsaâ„¢'*'

* = /.(*) + x=0 <M*>. ,./.(*) *Â«*Â«*<â€¢>â€¢ TSiâ€” â€¢

<p x (x) being assumed to be positive in the case of the last limit.

For exceptional cases, see Chapter XII.

STANDARD FORMS 9

7. Four important limits.

The following limits are important and will be frequently used

in this chapter : â€”

(I) L â„¢ (l + y- l == ^ whatever n may be.

(II) JJjtt+oU*

(III) Â«Â»Â»j ? i =1 â€ž g ,â€ž.

ryyv Lim SUl t _ -|

We proceed now to prove the four results given above : â€”

(I) As | Â£ | is nearly zero and is consequently less than 1, we can expand

(1 + t) n by the Binomial Theorem. Therefore

a +tr =i + nt + ^ n ^ + n{n '- 1 ^ n - 2 ^ + ....

Hence

(l + t)Â»-l |n(n-l) n(n~l)(n 2) i

â€” =n + ty 2! + 3! t+ . . . I.

But the numerical value of

fw(w-l) n(n-l )(n-2) i

\ 2! 3> t + ' ")

remains less than a finite quantity as t tends to zero. Therefore

lim f n(n-l) w(n-l ) (n-2) , j

Â£ = 0*1 2! 3! * â– ' * i '

and, consequently,

lim(l + 0' l -l

, = 1 =n.

i

(II) We have to prove that (1 + 1) 1 tends to e when t tends to zero by

assuming positive values, as well as when t tends to zero by assuming negative

values.

Case 1. t remains positive.

For each value of t, we can find two integers n, n + 1, such that

1

Thus

n n + 1

10 DIFFEEENTIAL CALCULUS

and

But

and

\ n) \ n) \ n) %

(i + jl y=(i + J lv +1 i .

V n + lj \ n + lj 1 | 1

n + 1

Therefore we have from (1)

n + 1

(1 \ m

1 + â€” 1 tends to e as

m /

the integer ?;i becomes greater and greater. Therefore fl + - J and

(1 V+ 1

1 + i both tend to e as the integer n becomes greater and greater ;

n+1/

also it is obvious that (1+ ) and â– both tend to 1. Hence it

n+1

i

follows from (2) that (1 + t) l tends to e as t tends to zero.

Case 2. t remains negative.

PuU=-v. Then

i

(1-

= (l + w)' c (l + w)

i)

where

Now when t tends to zero by assuming negative values, v and, consequently,

w tend to zero by assuming positive values. Therefore it follows from Case 1,

i i

that (1 + w)' r tends to the limit e. Hence (1 + 1) l tends to the limit e.

(Ill) By the Exponential Theorem,

Â«> s uih 6 . + p ( i *Â«)' t g(y t .

2 ! 3 !

STANDARD FORMS

11

Therefore

But the numerical value of

â€¢ }

f (logÂ«g) 2 *(log g g) 3

12! 3!

remains less than a finite quantity as t tends to zero. Therefore

lim / (log, a)* t(\og e af \ =Q

Ml 21 81 ' ' ' J

and, consequently,

lim a 1 â€” !

t = t

= log e a.

(IV) Describe a circle of unit radius and construct as in the adjoined

figure. Then

PM<PA<NA.

Therefore, for < t < , we have

sin t < t < tan t,

. -U-J-.

sin t cos t

Similarly it is proved that (1) holds also for

0>t>-l

i.e., 1<

- (1).

But it is obvious that hm rt -i- - 1.

t = cos t

Hence

, = 1 >

it follows from (1) that

lim t

* = s i n $

. m lim sin t -.

â€¢**Â«-o-r" L

8. Differential coefficient of x\

If /(a?) =# n , then f(x + fc) = (a? + fc)* and

W* lim (a? + ft)Â»-sÂ»

/w- fc=s0 a :

Two cases arise.

Case I. x^:0.

h\*

Fig. 1.

(x+hy-xÂ» =xn b ( 1+ 3 "L ^-i . ( 1+ *) ~ 1

In li h

h

x

12 DIFFERENTIAL CALCULUS

Now as h tends to zero, - also tends to zero. Therefore

x

â– /..or

x f lim (l + Q"-l l

- x \ t =o r ~ I â–

But by (I) of Art. 7,

l im (i + < )Â»_i

Therefore /'(x)=wx"- 1 , i.e., ^=nx"-'.

Case II. x = 0.

When n<0, /'(0) is non-existent, for/(0) has no meaning.

When n >

.,,â€ž> lim hÂ»_ lim .â€ž.,

Hence, if n < 1, /'(0) is oo or non-existent according as h n changes, or does not

change, its sign with h ; if n^ 1, /'(0) is 1 or according as n- 1 or > 1.

9. Differential coefficient of a*.

If/(#)=a*, then f(x + h)=a x+h , and

lim a x+ *â€” a x

/(*) =

/i=0 ft

lim a' 4 â€” 1

But by (III) of Art. 7,

~ a â€¢/*=() ~7T-

lim a*â€” 1 . â€ž

fc-0 A =1 Â° g ' a -

Therefore f'(x)=a x log e a, i.e., ?^-'==a r log, a.

STANDARD FORMS 13

EXAMPLES.

1. Write down the differential coefficients of x, x*, x*, ar T , x~*, 2 r , (i) T , e r >

e- r , 10*.

2. If f(x) = 2* 7 find f'(x).

7* = 0l h(2x + h) hi'

As ft tends to zero, h(2x + ft) tends to zero. Therefore

Hm 2 ft(2 *+* ) -l =; lim ff-l,

ft = 0~ft(2z + ft) Â£ = * ~ l0 & 2 -

Also evidently

Therefore / '(x) = 2*" x 2a; x log e 2.

10. Differential coefficient of sin x.

If /(#)=sin x, then f(x + h)= sin (# + /&), and

fi(â€ž\â€” lim sm fo + fe)â€” sin #

= lim^os(^^

; l= o JJ

2

lim

cos (#-{ - ) =cos x

But evidently

Io, by (IV) of Art. 7,

Therefore/'(^) = cos^ i.e.,â€” ( 4 ?L ^=cosx.

dx

lim 2__ lim sin t +

/i=o"T""~^=o * :

14 DIFFERENTIAL CALCULUS

11. Differential coefficient of cos x.

If f(x)=cos x, then f(x + h) = cos (x + h), and

,,, N lim cos (x + h)â€” cos x

o â€¢ ( , h\ â– h

h=0 S

lim J . / , *\

=â€” sin a? ;

d (cos x)

i.e., * , ; =â€” sin x.

dx

EXAMPLES.

1. Find the differential coefficient of sin 2x.

2. If f(x)= sin 2 a, find /'(a).

, u x lim sin 2 (x + h) â€” sin 2 x

f{x)= h=o â€” h~

_ lim sin (2x + h ) sin h

~h=0~ h

= sin 2x.

12. Differential coefficient of tan x.

If f(x) =tan x, then f(x + /i) = tan (# + /z), and

r l( x__ lim tan (# + ^) â€” tan x

jw- h=0 - h

___ lim sin (x + fo) cos # â€” sin # cos (a?4-/t)

/i=0 h cos (# + /&) cos a;

â€” ^ m sin h 1

h=0 h cos (x + h) cos x cos 2 # '

d (tan x) â€ž o

i.e., v , / =sec 2 x.

dx

13. Differential coefficient of cot x.

Proceeding as in the last article, we find that

d(cotx) = _ cosec2x

dx

STANDARD FORMS

14. Differential coefficient of sec x.

If/(#)=sec x, then f(x + h)-= sec (x + h), and

, H x lim sec (x + h)â€” sec x

_ lim cos #â€” cos (# + /&)

~~ /*,=() /& cos (# + /&) cos a;

OP

DIFEEBENTIAL CALCULUS

A TEXT-BOOK

OF

DIFFERENTIAL CALCULUS

WITH NUMEROUS WORKED OUT EXAMPLES

BY

GANESH^^RASAD

B.A. (Cantab.), D.Sc. (Allahabad)

MEMBER OP THE LONDON MATHEMATICAL. SOCIETY, OF THE DEUTSCHE

MATHEMATIKER-VEREINIGUNG, OP THE CTRCOLO MATEMATICO DI PALERMO, ETC.

FELLOW OF THE UNIVERSITY OF ALLAHABAD

AND PROFESSOR OF MATHEMATICS IN QUEEN'S COLLEGE, BENARE8

LONGMANS, GEEEN, AND CO.

39 PATERNOSTER ROW, LONDON

NEW YORK, BOMBAY, AND CALCUTTA

1909

All rights reserved

OfV f

^

PREFACE.

In this work it has been my aim to lay before students a

strictly rigorous and, at the same time, simple exposition of

the Differential Calculus and its chief applications. The

present volume is intended for beginners and is so designed

as to meet the requirements of Part I. of the Cambridge

Mathematical Tripos Examination, and of the Examinations

for the B.A. and B.Sc. degrees of Indian Universities.

The chief characteristics of the present work may be

indicated as follows : â€” (1) The fundamental principles of the

Differential Calculus have been based on a purely arith-

metical foundation. Thus, the various theorems have been

carefully enunciated and their proofs have been made quite

independent of geometrical intuition. In this connection,

I may specially mention the chapters on Bolle's Theorem

and Taylor's Theorem, Maxima and Minima, and Indeter-

minate Forms. (2) Almost every article is followed by

worked out examples, specially suited for illustrating the

article. There are also numerous exercises in every chapter.

(3) A special chapter deals with curve-tracing and the im-

portant properties of the best- known curves. (4) The order

in which the chapters are arranged is intended to enable the

beginner to study the simple geometrical applications of the

Differential Calculus immediately after he has learnt differen-

tiation. (5) The miscellaneous notes A and B are intended

r- O t i* f> O

vi PREFACE

to give the ambitious student a glimpse of the modern

researches in the Differential Calculus.

This volume is based on my experience in teaching the

elements of the Differential Calculus to a large number of

pupils. It is, therefore, throughout of an elementary

character. But, as certain parts of it may be found difficult

by beginners, they have been marked with an asterisk and

may be omitted in a first reading.

A few words may be said here about Chapter I. In a

mathematical book, which professes to be rigorous in its

treatment, it is essential that the definitions be carefully

worded. This has been done in the present work. It is,

however, possible that, for this reason, Chapter I. may be

found heavy reading by some students. To them my advice

is this : If you do not grasp the full meaning of a definition

in a first reading, leave it and after reading Chapters II. and

III. come back to Chapter I., and then you will understand

the definition better.

In writing the present volume, I have derived much help

from two books, viz. the excellent little manual, ' Calcolo

Differenziale ' of Professor Ernesto Pascal, and Todhunter's

1 Treatise on the Differential Calculus.' For the historical notes

in Chapter VIII., as well as for most of the examples on that

chapter, I am indebted to Professor Gino Loria's ' Spezielle

algebraische und transscendente ebene Kurven.' Of the re-

maining examples in this volume, a large number are common

to all English text-books on the subject, some are original and

the others are taken from the Tripos Examinations of recent

years ; in the case of the more important examples belonging

to the last category, the sources are cited in the text.

To ray friend and former pupil, Mr. Lakshmi Narayan,

M. A., Professor of Mathematics at the Central Hindu College,

PEE FACE vii

Benares, I am much indebted for some valuable suggestions

and for assistance in revising the proof-sheets. I am also

indebted to my friend Dr. S. C. Bagchi, B.A. (Cantab.), LL.D.

(Dublin), Principal of the University Law College of Calcutta,

whose criticisms and suggestions, relating to Chapters V.,

VIII., and XL, have materially enhanced the usefulness of the

present volume.

G. PBASAD.

Benares, July 1909.

CONTENTS.

[N.B. â€” The portions marked with an asterisk may be omitted in a first reading.]

CHAPTEE I.

Definitions.

ART. PAGE

1. Variable 1

2. Function 1

3. Limit 2

4. Continuity 4

5. Differential coefficient 5

*Examples on Chapter 1 6

9.

10.

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

"

CHAPTER II.

Standard Forms.

Introductory

Four important limits

Differential coefficient of x H

Differential coefficient of a x

Differential coefficient of sin x

Differential coefficient of cos x

Differential coefficient of tan x

Differential coefficient of cot x

Differential coefficient of sec x

Differential coefficient of cosec x

Differential coefficient of vers x

Differential coefficient of log a x

Differential coefficient of sin -1 x

Differential coefficient of cos -1 x

Differential coefficient of tan -1 x and cot -1 x

Differential coefficient of sec -1 x and cosec -1 x

Differential coefficient of vers -1 x

Table of results to be committed to memory

Examples on Chapter II

8

9

11

12

13

14

14

14

15

15

15

16

17

18

19

19

20

21

22

x CONTENTS

CHAPTER in.

Fundamental Rules for Differentiation.

ART. PAGE

24. Constant 24

25. Product of constant and function 24

26. Sum of two functions 24

27. Product of two functions 25

28. Product of more than two functions 25

29. Quotient of two functions 27

30. Function of a function 28

31. Inverse functions , . . 28

32. The form \4>{x)}W> 29

Examples on Chapter III. 30

CHAPTER IV.

Tangents and Normals.

33. Tangent : Definition and Cartesian equation .

34.

35.

36.

37.

38.

39.

40.'

41.

42.

43.

44.

45.'

46.

47.

ds

v- sv

Note on the equation ^r

Normal : Definition and Cartesian equation

Cartesian subtangent and subnormal ....

Polar coordinates. Angle between tangent and radius vector

ds /â€ž, . /dr\ 2

\de)

rÂ»+

Note on the equation â€”-= . /

Polar subtangent and subnormal

Perpendicular from pole on tangent .

Pedal equation

Inversion. Pedal curves. Polar reciprocals

Examples on Chapter IV

35

36

39

40

41

42

43

44

45

46

47

CHAPTER V.

Asymptotes.

Definition of asymptote 52

General rule for finding asymptotes from Cartesian equation . . 53

Parallel asymptotes 56

Asymptotes by inspection 57

Note on curvilinear asymptotes 58

General rule for finding asymptotes from polar equation ... 59

Note on circular asymptotes 59

Examples on Chapter V 60

CHAPTER VI.

Curvature.

Centre of curvature : Definition and Cartesian coordinates

Radius of curvature : Definition and formulae .

63

66

STENTS xi

ART. PAOT

48. Circle and chord of curvature 70

49. Evolute 71

50. Concavity and convexity. Point of inflexion 73

Examples on Chapter VI 75

CHAPTER VII.

- Envelopes.

51. Family of curves 78

52. Definition of envelope 78

53. Rule for finding the envelope of a family of straight lines . . 7&

54. General rule 81

Examples on Chapter VII. 82

â™¦CHAPTER VIII.

Curve Tracing. Properties of Special Curves.

55. Introductory 84

56. Rules for Cartesian equations 84

Note on multiple points 85

57. Rules for polar equations 87

58. Parabola. Ellipse. Hyperbola 89

59. Semi-cubical parabola. Cissoid. Folium 91

60. Lemniscate. Cardioid. Conchoid 92

61. Cycloid. Catenary. Tractrix . . 93

62. Logarithmic spiral. Archimedian spiral. Sine spiral ... 95

Examples on Chapter VIII. . . â– 96

CHAPTER IX.

Successive Differentiation.

63. Definitions 99.

64. Standard results 100

65. Leibnitz's theorem 102

66. Use of partial fractions. Recurrence formulas 103

Examples on Chapter IX 107

CHAPTER X.

Rolle's Theorem and Taylor's Theorem.

67. Rolle's theorem. The theorem of the mean value . . . .110'

68. Taylor's development in finite form 112

69. Taylor's theorem. Maclaurin's theorem 115

*Note on contact of curves . 117

Examples on Chapter X H^

xii CONTENTS

CHAPTER XI.

Maxima and Minima.

ART. PA'

70. Definitions 121

71.* Two theorems ". 122

72. General rule for finding maxima and minima 125

Examples on Chapter XI ; 127

CHAPTER XII.

Indeterminate Forms.

73. Introductory 128

74. Cauchy's theorem. The fundamental form - 130

75. The Forms ~ , x Â°o co _ co oÂ°, 1Â°Â°, coo 132

CO

Note on compound indeterminate forms 135

Note on infinitesimals and infinities 137

Examples on Chapter XII. 138

MISCELLANEOUS NOTES.

A. Weierstrass's function 140

B. Rolle's theorem and Taylor's theorem , 143

C. Partial differentiation 146

\J-

DIFFERENTIAL CALCULUS.

CHAPTEE I.

DEFINITIONS.

1. Variable. Let # be a symbol which takes successively every

numerical value from a given number a to another given number /3.

Then x is called a variable and the totality of the values of x

constitutes the domain of x.

We will represent the domain of x by the symbol (a, fi).

Note. If k be a number, it will be convenient to use the symbol | k \

to denote the absolute value of k, i.e., the value of k without regard to its

sign. Thus

I -2 I = I 2-1 = 2.

2. Function. By a function of x, denned for a given domain,

is understood a quantity which has a single and definite value for

every value of x in its domain.

We generally denote functions of x by such symbols as f(x),

*(*), F(x), f(x).

EXAMPLES.

1. # 2 , 2 X , sin x are functions of x whatever be the domain

of x. But sin -1 x cannot be a function of x for such a domain

as (2, 3).

2. The temperature curve at a certain place is y = 80 + 10 sin x. If the

temperatures recorded are all different, the highest and lowest temperatures

being 90Â° and 80Â° respectively, find the domain of x.

3. For the domain (0, 1), a function may be defined by saying that it is

x 11

zero or - according asÂ£C = 0or->#> , n having the values 1 2, 3, etc.

n nâ€” n+l

2 Â»>V DIFFERENTIAL CALCULUS

4. For .the -domain (#, -1)[, & function f(x) may be defined by saying that

f(x) is 2 or 3 according as x is rational or irrational.

3. Limit. A and a being both finite, A is said to be the

limit of f(x) for #=a if, for any number 8, however small but

greater than zero, there exists a corresponding number c>0

such that

x having every value such that

0<

<8,

<â‚¬.

Note 1. This definition may be expressed in a different form, viz. : A is

said to be the limit of f(x) for x = a, if f(x) differs from A by less than any

assigned quantity, however small, when x has any value sufficiently near to a.

We will use the notation __ f(x) to denote the limit of f(x)

for x=a.

f(x) is said to be oo, if, for any positive number N however large, there

x = a

exists a corresponding number e > such that f(x) > N, x having every value

such that

0< 05-a <â‚¬.

f(x) = A, if, for any number 5, however small but greater than zero,

there exists a corresponding number N>0 such that

U-/(a?) I <8,

x having every value greater than N.

li

Definitions similar to the above hold for the cases when fix)- -oo,

x = a

lim f(x) = A lita /(*)=oo,etc.

Xâ€” â€” QO X = QO

Note 2. The notion of limit, on which Differential Calculus is based, is

not so unfamiliar to the beginner as he might at first imagine. For, in his

algebraic studies he must have become acquainted with this notion in connec-

tion with the sums of infinite series. For example, what is meant by saying

that 2 is the sum of the series

i + 2+ 2 \+23 + â€¢ â– â€¢ toinfinit y ?

Nothing but this : 2 is the limit to which S,â€ž the sum to n terms, tends as

n is made greater and greater.

DEFINITIONS

EXAMPLES.

1. **"" x 2 is a 2 . For, take any number Â£ however small, but

Lim

greater than zero. Now, if

#â€” a

<Â«,

where

< 1 ; hence

x=a + 6e,

x + a=2a + 0e,

and, consequently,

Therefore, since

x + a J <2

+ *.

x 2 â€”a

2 =(x-

-a)(x+a),

x 2 â€”a 2

<e

x+a <el% a

Therefore, if e is such that

.(,

a 1 + Â£ )Â±S,

x 2 -a 2 1 <a

for every value of x satisfying the condition

xâ€”a

<â‚¬.

And such a value of e is any positive number which is equal to or

less than the positive root of

tU I a I + A-S=0;

for, as the product of the roots of this equation is â€”2, a negative

quantity, one of the roots is positive and the other negative.

2. In the last example, prove that, when 8 is taken to be â€” , e may be

taken to be

2 | a | 10"

3. Prove that hm cosz = l.

x =

4. If _ f(x) exists and is finite, it follows that, for any number 5, how-

b2

4 DIFFEBENTIAL CALCULUS

ever small but greater than zero, there exists a corresponding number â‚¬ >

such that

|/(z,)-/(x 2 )| <5

for every pair x v x 2 satisfying the conditions

0< a?,â€” a < â‚¬ , 0< \ x 2 â€” a\ <â‚¬.

5. Prove that im cos is non-existent. Let x, and x 2 be respectively

x = x

equal to â€” and ; r- t n being an integer. Then

* 2mr (2n + l)?r

cos â€” cos = 2,

x x x 2

however large n may be. Hence it follows that in * cos is non-existent.

x = x

For, if this limit existed, according to the preceding example, it would be

possible to find a value of ft so large that

cos cos â€” < 5

even when 5 < 2.

6. Prove that im rt , is non-existent.

x = L

2 + e x

4. Continuity. A function f(x) is said to be continuous for

x=a, if im J(x) exists, is finite and equals f(a).

x=a

EXAMPLES.

1. x 1 is continuous for #=a. For, x 2 exists and equals a 2 .

x=a ^

2. A function f(x) is defined by saying that it equals 1 or

e~*\ according as x is zero or different from zero. This function

is discontinuous for #=0. For, e x ' 1 is zero and is not equal

to /(0).

3. If f(x) be continuous for x = 0, prove that /(0) must be zero, when

f(x) = x sin - for values of x different from zero.

4. What are the points of discontinuity of the function given in Ex. 3 of

Art. 2 ?

DEFINITIONS 5

5. Differential Coefficient. By the differential coefficient of a

function f{x) for x=a is understood

lim /(s)-/(a) ^ lim /(a + ft)-/(a)

#=a # - a ' ' 7&=0 /&

The differential coefficient of f(x) for #=& will be denoted by the

symbol f(a) ; the differential coefficient, considered as a function

df(x)

of x, will be generally represented by -} orf'(x).

ctx

Note 1. The process of finding/' (x) is called differentiating f(x).

Note 2. For the geometrical meaning of f'(x), see Art. 33, Note 1.

Note 3. The beginner should not think that â€¢} ' means the ratio of

ax

df(x) to dx. To think so would be as wrong as to think that sin x means the

product of sin and x. Just as sin is meaningless, so are df(x) and dx. As

has been already stated, the symbol *} ' stands for

dx

lim f(x + h) -f(x)

h = Q h

This notation was introduced by Leibnitz in 1676.

EXAMPLES.

1. Find the differential coefficient of x 6 for x=-l.

Here

xâ€”1 xâ€”1

Therefore

/'(1)= Km /(*)-Al) = lim (a .* + s+l)=3.

x=l xâ€”1 X=I

2. Find *&.

dx

3. A function f(x) equals zero or x cos -, according

zero or different from zero. Prove that/'(0

Here

rio)= lim SMSM

J v ' x=0 x

lim 1

#=0 x

But, by Ex. 5, Art. 3, lm * cos - is non-existent. Therefore /'(())

x=0 x J v '

is non-existent.

as x is

x

stan-(l).

6 DIFFEBENTIAL CALCULUS

4. A function f(x) is denned by saying that it equals 0, x or - x according

as a: is 0, > or <0. Prove that/'(0) does not exist.

j

5. Find the differential coefficient of x 3 for x = 0.

6. If f(x) has the same value whatever x may be, prove that

/'(*)-0.

Here

f'tmX- Hm / ( a? + ft )-/( a? ) = lira 9.

1 v ; fc=0 & *=0 ft

â™¦Examples on Chapter I.

1. If /(*)Â»- h â„¢ tan" 1 ( x \ prove that f(x) is equal to 0, 1 or -1

according as x is 0, > or < 0.

2. Trace the curve

y= l[m

3. If /(a?) = , prove that f(x) is or 1 according as a; is or

different from 0.

4. Give the graph of

lim (l + simro;)' 1 â€” 1

y a Â» * ,

n- co (1 + sinirx) w + 1

5. If <Â£>(#) s - â€” â€” p prove that the limit to which <f>(sin n ! *#) tends

when the integer n is made larger and larger, is or 1 according as x is

rational or irrational.

6. A function f(x) is denned by saying that it equals 0. or sin (- )

\sin xj

according as x is, or is not, a multiple of Â». Find the points of discontinuity

of /(*).

7. Trace the curve

y* lim (simp)*- 1 .

n = co

8. A function f(x) is denned by saying that it equals or sin / \

l Sin xj

according as x is either zero or a submultiple of , or neither of these. Prove

IT

that there are an infinite number of points of discontinuity of f(x) between

a and |3, where a < < 3.

DEFINITIONS 7

9. If f(x) = or according as x is zero or different from zero, trace

l-e x

the curve y =f{x).

10. Trace the curve

li m x 2n sin - +x 2

find the values of y at the points x Â». Â± 1, and discuss whether 2/ is continuous

at these points.

[Math. Tripos, 1901.]

. 1

sin ~

11. A function f(x) is defined by saying that it equals or Â£ according

log x*

as x is zero or different from zero. Has f(x) a differential coefficient for x = ?

12. In each of the following cases discuss the question of the existence of

the differential coefficient for x = a : â€”

_ _i

(i) f(x) = e ( * -a)a whenx^a,

/(#) = when x = a.

(ii) /(#) = (# -a) 2 cos when xÂ±a,

x-a '

/(#) s when sc = a.

(iii) /(x) = <c - a when #>a,

/(ic) = a-cc when x < a.

(iv) f(x) = (x â€” a) cos when #=j=a,

/(#) = when x = a.

[Calcutta Univ., 1906.]

CHAPTEE II.

STANDARD FORMS.

6. Introductory. It is the object of the present chapter to

investigate and tabulate the results of differentiating the simple

elementary functions, viz., x n , a x , sin x, cos x, tan x, cot x, sec x,

cosec x, vers x } \og a x, sin -1 x, cos -1 x, tan -1 x, cot -1 x, sec -1 x,

cosec -1 x, vers -1 x.

It will be seen later on that, by means of certain rules to be

given in Chapter III. and a knowledge of the standard forms of

the present chapter, most of the ordinary functions can be easily

differentiated.

Throughout the book we shall always consider the inverse

functions to be so denned that

â€” -j<sin -1 # _<-, CKcos -1 #Â£7r,

A . A

- 2 ^tan *x < 2 ,- 2 ^_cot l *<^

Oj^sec -1 a? <Â»,â€” ^cosec" 1 #_S^,

0<yQr$~ l x<j.

Note. Throughout this book we shall take for granted the truth of the

following theorem : In general,

1Z { *Â« * m) . Â£ { *i - ** } â– x t { :;g } â– Â£ { <^h }

are respectively equal to

lim

lim . , , , lim . ,, lim . , * lim . , v scrsaâ„¢'*'

* = /.(*) + x=0 <M*>. ,./.(*) *Â«*Â«*<â€¢>â€¢ TSiâ€” â€¢

<p x (x) being assumed to be positive in the case of the last limit.

For exceptional cases, see Chapter XII.

STANDARD FORMS 9

7. Four important limits.

The following limits are important and will be frequently used

in this chapter : â€”

(I) L â„¢ (l + y- l == ^ whatever n may be.

(II) JJjtt+oU*

(III) Â«Â»Â»j ? i =1 â€ž g ,â€ž.

ryyv Lim SUl t _ -|

We proceed now to prove the four results given above : â€”

(I) As | Â£ | is nearly zero and is consequently less than 1, we can expand

(1 + t) n by the Binomial Theorem. Therefore

a +tr =i + nt + ^ n ^ + n{n '- 1 ^ n - 2 ^ + ....

Hence

(l + t)Â»-l |n(n-l) n(n~l)(n 2) i

â€” =n + ty 2! + 3! t+ . . . I.

But the numerical value of

fw(w-l) n(n-l )(n-2) i

\ 2! 3> t + ' ")

remains less than a finite quantity as t tends to zero. Therefore

lim f n(n-l) w(n-l ) (n-2) , j

Â£ = 0*1 2! 3! * â– ' * i '

and, consequently,

lim(l + 0' l -l

, = 1 =n.

i

(II) We have to prove that (1 + 1) 1 tends to e when t tends to zero by

assuming positive values, as well as when t tends to zero by assuming negative

values.

Case 1. t remains positive.

For each value of t, we can find two integers n, n + 1, such that

1

Thus

n n + 1

10 DIFFEEENTIAL CALCULUS

and

But

and

\ n) \ n) \ n) %

(i + jl y=(i + J lv +1 i .

V n + lj \ n + lj 1 | 1

n + 1

Therefore we have from (1)

n + 1

(1 \ m

1 + â€” 1 tends to e as

m /

the integer ?;i becomes greater and greater. Therefore fl + - J and

(1 V+ 1

1 + i both tend to e as the integer n becomes greater and greater ;

n+1/

also it is obvious that (1+ ) and â– both tend to 1. Hence it

n+1

i

follows from (2) that (1 + t) l tends to e as t tends to zero.

Case 2. t remains negative.

PuU=-v. Then

i

(1-

= (l + w)' c (l + w)

i)

where

Now when t tends to zero by assuming negative values, v and, consequently,

w tend to zero by assuming positive values. Therefore it follows from Case 1,

i i

that (1 + w)' r tends to the limit e. Hence (1 + 1) l tends to the limit e.

(Ill) By the Exponential Theorem,

Â«> s uih 6 . + p ( i *Â«)' t g(y t .

2 ! 3 !

STANDARD FORMS

11

Therefore

But the numerical value of

â€¢ }

f (logÂ«g) 2 *(log g g) 3

12! 3!

remains less than a finite quantity as t tends to zero. Therefore

lim / (log, a)* t(\og e af \ =Q

Ml 21 81 ' ' ' J

and, consequently,

lim a 1 â€” !

t = t

= log e a.

(IV) Describe a circle of unit radius and construct as in the adjoined

figure. Then

PM<PA<NA.

Therefore, for < t < , we have

sin t < t < tan t,

. -U-J-.

sin t cos t

Similarly it is proved that (1) holds also for

0>t>-l

i.e., 1<

- (1).

But it is obvious that hm rt -i- - 1.

t = cos t

Hence

, = 1 >

it follows from (1) that

lim t

* = s i n $

. m lim sin t -.

â€¢**Â«-o-r" L

8. Differential coefficient of x\

If /(a?) =# n , then f(x + fc) = (a? + fc)* and

W* lim (a? + ft)Â»-sÂ»

/w- fc=s0 a :

Two cases arise.

Case I. x^:0.

h\*

Fig. 1.

(x+hy-xÂ» =xn b ( 1+ 3 "L ^-i . ( 1+ *) ~ 1

In li h

h

x

12 DIFFERENTIAL CALCULUS

Now as h tends to zero, - also tends to zero. Therefore

x

â– /..or

x f lim (l + Q"-l l

- x \ t =o r ~ I â–

But by (I) of Art. 7,

l im (i + < )Â»_i

Therefore /'(x)=wx"- 1 , i.e., ^=nx"-'.

Case II. x = 0.

When n<0, /'(0) is non-existent, for/(0) has no meaning.

When n >

.,,â€ž> lim hÂ»_ lim .â€ž.,

Hence, if n < 1, /'(0) is oo or non-existent according as h n changes, or does not

change, its sign with h ; if n^ 1, /'(0) is 1 or according as n- 1 or > 1.

9. Differential coefficient of a*.

If/(#)=a*, then f(x + h)=a x+h , and

lim a x+ *â€” a x

/(*) =

/i=0 ft

lim a' 4 â€” 1

But by (III) of Art. 7,

~ a â€¢/*=() ~7T-

lim a*â€” 1 . â€ž

fc-0 A =1 Â° g ' a -

Therefore f'(x)=a x log e a, i.e., ?^-'==a r log, a.

STANDARD FORMS 13

EXAMPLES.

1. Write down the differential coefficients of x, x*, x*, ar T , x~*, 2 r , (i) T , e r >

e- r , 10*.

2. If f(x) = 2* 7 find f'(x).

7* = 0l h(2x + h) hi'

As ft tends to zero, h(2x + ft) tends to zero. Therefore

Hm 2 ft(2 *+* ) -l =; lim ff-l,

ft = 0~ft(2z + ft) Â£ = * ~ l0 & 2 -

Also evidently

Therefore / '(x) = 2*" x 2a; x log e 2.

10. Differential coefficient of sin x.

If /(#)=sin x, then f(x + h)= sin (# + /&), and

fi(â€ž\â€” lim sm fo + fe)â€” sin #

= lim^os(^^

; l= o JJ

2

lim

cos (#-{ - ) =cos x

But evidently

Io, by (IV) of Art. 7,

Therefore/'(^) = cos^ i.e.,â€” ( 4 ?L ^=cosx.

dx

lim 2__ lim sin t +

/i=o"T""~^=o * :

14 DIFFERENTIAL CALCULUS

11. Differential coefficient of cos x.

If f(x)=cos x, then f(x + h) = cos (x + h), and

,,, N lim cos (x + h)â€” cos x

o â€¢ ( , h\ â– h

h=0 S

lim J . / , *\

=â€” sin a? ;

d (cos x)

i.e., * , ; =â€” sin x.

dx

EXAMPLES.

1. Find the differential coefficient of sin 2x.

2. If f(x)= sin 2 a, find /'(a).

, u x lim sin 2 (x + h) â€” sin 2 x

f{x)= h=o â€” h~

_ lim sin (2x + h ) sin h

~h=0~ h

= sin 2x.

12. Differential coefficient of tan x.

If f(x) =tan x, then f(x + /i) = tan (# + /z), and

r l( x__ lim tan (# + ^) â€” tan x

jw- h=0 - h

___ lim sin (x + fo) cos # â€” sin # cos (a?4-/t)

/i=0 h cos (# + /&) cos a;

â€” ^ m sin h 1

h=0 h cos (x + h) cos x cos 2 # '

d (tan x) â€ž o

i.e., v , / =sec 2 x.

dx

13. Differential coefficient of cot x.

Proceeding as in the last article, we find that

d(cotx) = _ cosec2x

dx

STANDARD FORMS

14. Differential coefficient of sec x.

If/(#)=sec x, then f(x + h)-= sec (x + h), and

, H x lim sec (x + h)â€” sec x

_ lim cos #â€” cos (# + /&)

~~ /*,=() /& cos (# + /&) cos a;

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